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I have real-world data, which after analysis, produces a series of curves. The data has noise, but each vector $u_1(x), u_2(x), \ldots, u_6(x)$ fits extremely well to a polynomial of that same order, eg. $u_3$ is very well fit by a cubic polynomial.

In addition to this, the polynomials are orthonormal-ish over the valid range of the function. That is

$$ \int_0^{1/2} u_i(x) u_j(x) \ dx = c \delta_{ij} $$

with the same constant $c$ for all values of $i=j$. In addition, the function is not quite zero when $i \neq j$ but small, about $.05c$. This may be due to numerical imprecision or the wrong integrated interval.

See the picture below of the data with the fits superimposed:

enter image description here

Questions: If I have a hypothesis that the polynomials form a orthonormal basis set, how can I show this (other than what I've done)? More importantly, what does this tell me? Is there a "standard-form" that I can convert my polynomials to check if they fall under any of the classic orthogonal polynomials?

Edit: I've checked numerically, between each interval of the roots of $u_i$ lies exactly one root of the polynomial $u_{i+1}$, so this property of orthogonal polynomials seem to hold.

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  • $\begingroup$ Do you happen to have the explicit expressions for the $u_k$ around? $\endgroup$ Commented Apr 6, 2013 at 13:15

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Two signals are called orthonormal if they are both orthogonal, namely their all pairwise innerproducts results in $0$ (when $i\neq j$), except for the same signal itself (when $i= j$), and unit vector. This says that their inner product should be $c\cdot \delta$, where $c=1.$

When we come to your question, you can either show this in the discrete domain via using summations and inner products, or with less accuracy using some polinomial approximations in the continous domain as you did.

If your data follows some orthogonal polinomial structure, the meaning of this can vary from one application to the other. The good property of this structure is that these signals dont interfere each other if they are transmitted simultaneously in a communications scenario, where multiple access is of primary interest (see http://en.wikipedia.org/wiki/Orthogonal_frequency-division_multiplexing) or when the channel is severe and the signal frequency is desired to be spread to the whole frequency band to be robust to signal interference or fading etc.. (pls see http://en.wikipedia.org/wiki/Orthogonal_frequency-division_multiplexing). They are also used in biomedical signal processing, for example zernike polinomials are used to model the eye structure. All and all, according to my best knowledge, they provide two basic properties

$1$: Robustness or robust modeling of the problem under consideration.

$2$: Efficiency in terms of sharing; as in CDMA systems.

One can of course check the data that you obtained for some classical orthogonal polinomials using a criterion such as minimum mean squared error. In this case I would suggest to check firstly these two polinomials:

http://en.wikipedia.org/wiki/Legendre_polynomials

http://en.wikipedia.org/wiki/Krawtchouk_polynomials

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